Transfer Operators on Complex Hyperbolic Spaces

Let B n be the unit ball in the n-dimensional complex space and let ∆ be the Bergman Laplacian on it. For ∈ C such that |ℜ(i )| < n we give explicitly the transfer operator from the space of holomorphic functions B n onto an eigenspace E + (B n ) of ∆. This answers a question raised by Eymard in (2). As application, for = −i with 0 < < n , we get that the classical Hardy space H 2 (B n ) is isometrically isomorphic to the space